Prime numbers, prime factors and composite numbers

In a nutshell

Prime and composite numbers are special categories of numbers based on how many factors they have. There are $25$ prime numbers and $74$ composite numbers between $1$ and $100$ .

Prime numbers and composite numbers

Prime numbers only have two factors, $1$ and itself. Composite numbers have more than two factors.

Example 1

Identify whether the following numbers are prime or composite: $2,10,13,25$ .

Number	Factors	Prime or Composite
$2$	$1,2$	$Prime$
$10$	$1,2,5,10$	$Composite$
$13$	$1,13$	$Prime$
$25$	$1,5,25$	$Composite$

Note: $1$ is neither a prime or composite number because it only has one factor, $1$ .

Prime factors

Factors of a number which are prime are prime factors. They can be found using a factor tree to breakdown a number into its prime factors which can be multiplied together to give the original number. This can be done by following the procedure below:

PROCEDURE

$1$	Write down the given number at the top.
$2$	Find any two numbers which multiply to give the top number.
$3$	Write down these two factors beneath the original number at the end of a left and right "branch".
$4$	Repeat this process for all the numbers in the tree until each "branch" ends in a prime factor.
$5$	Circle each prime factor, and write them all multiplied by each other.

Example 2

What is $32$ as a product of its prime factors?

Write $32$  on the top, and find two numbers that multiply to give $32$ :

$32=2\times16$

Write $2$ and $16$ beneath $32$ . $2$ is a prime number so circle it.

${\begin{aligned} &\space \space \,\,\,\,32\\\ &\,\,\swarrow\searrow& \\ &\textcircled{2} \quad16 \end{aligned}}$ 

As $16$ is not a prime number, find two numbers that multiply to give $16$ such as: $2 \times 8$ . Add this to the factor tree and circle $2$ since it is prime.

$\quad{\begin{aligned} &\space \space \,\,\,\,32\\\ &\,\,\swarrow\searrow& \\ &\textcircled{2} \quad16 \\ &\quad\,\,\swarrow\searrow\\ &\space\space\space\,\,\textcircled2 \,\,\quad8 \end{aligned}}$

As $8$ is not a prime number, find two numbers that multiply to give $8$ such as: $2 \times 4$ . Add this to the factor tree and circle $2$ since it is prime.

$\quad\quad{\begin{aligned} &\space \space \,\,\,\,32\\\ &\,\,\swarrow\searrow& \\ &\textcircled{2} \quad16 \\ &\quad\,\,\swarrow\searrow\\ &\space\space\space\,\,\textcircled2 \,\,\quad8 \\ &\quad\quad\,\,\,\swarrow\searrow \\ &\quad\space\space\,\,\textcircled2 \quad\space\,\,\, 4 \end{aligned}}$

As $4$ is not a prime number, find two numbers that multiply to give $4$  such as $2 \times 2$ . Add this to the factor tree and circle both $2$ s since they are prime.

$\quad\quad\quad\space{\begin{aligned} &\space \space \,\,\,\,32\\\ &\,\,\swarrow\searrow& \\ &\textcircled{2} \quad16 \\ &\quad\,\,\swarrow\searrow\\ &\space\space\space\,\,\textcircled2 \,\,\quad8 \\ &\quad\quad\,\,\,\swarrow\searrow \\ &\quad\space\space\,\,\textcircled2 \quad\space\,\,\, 4 \\ &\quad\quad\quad\space\,\,\swarrow\searrow \\ &\quad\quad\quad\textcircled2 \quad\space \,\,\textcircled2\end{aligned}}$

Only prime factors remain so the tree is complete.

Write $32$ as a product of its prime factors.

Therefore, $\underline{2 \times 2\times 2\times 2\times 2=32.}$

Note: $1$ is not a prime number so it does not appear in the factor tree.